I know you said you have a math background, but I'm gonna write as though you don't because I think this is a super interesting question which gets at the heart

I know you said you have a math background, but I'm gonna write as though you don't because I think this is a super interesting question which gets at the heart of something that confused me for a while, and it deserves a discussion a layman can understand. So the best way to think about numbers (in this situation. in this situation. obviously there are other ways to think about them. i'm not a platonist. nobody is a platonist enough to justify the amount of pedantry this take would normally inspire) is as abstract objects which exist in and of themselves. A particular base is just a way to write these abstract objects in a particular way. You can write numbers in a way which doesn't require a base, or is so annoying to describe as a base that you honestly shouldn't, such as the Roman system. This is important because there's a really fundamental sense in which 3 (decimal) and 10 (binary) are the same thing. When I write '3' I'm pointing you towards the abstract object |||. But that's just convention! Here on HN, I could write '1A' and expect many readers to see it as the same as '26' or '10011010'. With the naturals (and 0) you can think of 3 as fundamentally those three tallies, or three apples, or whatever. For the reals, algebraicists picture an element of a complete ordered field (my brain paints these, like all objects viewed abstractly, as pale tan circles) and analysts picture a point on a line. This representation, of course, doesn't scale, which is why we pick some number of symbols (usually 10, sometimes 16, 8, or 2, and in my very favorite dreams 12) and just agree to map them onto specific abstract objects. Then we can make all the other numbers in a super compact notation without making up any new symbols. So what about 0 and 1? Well 0 is the number which, when added to any x, gives x. And 1 is the number which, when multiplied with any x, gives x. Now I know what you're saying; "Ok, why do I care? 3 is the number of legs a table needs so it won't fall down, but it's not special". We see for any sane base that (so long as we're not just making up symbols) we'll write 0 just as 0 (since it's 0n^0 + 0n^1 + 0n^2...). What base is that expansion written in? I did it in decimal, but it doesn't matter! You can just as easily write (0n^zero + 0n^one + 0n^two...). Similarly, since it is just a fact about numbers that x^0=1 (or, if you like, x^zero=one), we'll write 1 as 1 (from 1n^0+0n^1...). If you trust that that relation holds up for irrational x, which I can't make you do but you definitely should, then you see that we'll also be writing 0 and 1 as 0 and 1 in base pi too. So what is pi^0? It's one! One in what base you ask? All of them! It's equal to the number which '1' represents. In a way that's because 0 is always 0 and 1 is always 1, but really it's the other way around. 0 is 0 and 1 is 1 regardless of base because pi^0, and 10^0, and any other number to the 0th power, is 1. Basically this exact issue is what made Tensor algebra so difficult for me, and I suspect a stumbling block for many people at some point. There you're talking about the basis of a vectorspace, rather than the base of a number system, but it's basically exactly the same thing. The key comes from thinking of objects floating in space with symbols pointing to them. We can operate on the symbols in all sorts of ways, and note cool patterns, but if you confuse them with the objects they point to you'll eventually be in for a bad time.